{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:f0e269d8-bae5-4032-9611-4e7c7b5f8518",
  "labels": {
    "zh-CN": "交换代数",
    "en": "commutative algebra"
  },
  "alternative_labels": [
    {
      "lang": "en",
      "value": "commutative"
    }
  ],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "1911778e27b5981aa5ea154808a5ddc5eebca3630b7656822291fbc24522c3f0",
  "occurrences": [
    {
      "concept_id": "urn:uuid:f0e269d8-bae5-4032-9611-4e7c7b5f8518",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "zh-CN",
      "route": "zh/chapter-06/01-algebras-and-ideals",
      "anchor": "concept-6-commutative-algebra",
      "chapter": "6",
      "section": "6.1",
      "offset": 2034,
      "range": [
        6790,
        7034
      ],
      "source_revision": "b14007c90d889a528ad167bc399531861f998294a0ab6687318a2d3cdc0b7deb",
      "source_category": "translation",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "交换代数(commutative algebra)",
      "context": {
        "range": [
          6208,
          7035
        ],
        "quote": "我们使用大写字母 $A$、$B$ 等来表示向量，因为代数最常出现在向量空间 $V$ 上的线性算符空间之中。然而，原则上没有什么能阻止使用更常见的记号 $u$、$v , . .$. 来表示向量，并写 $uv$ 表示它们的乘积。向量乘积一直用向量的简单并置来表示，但根据上下文，也可能出现其他记号，例如 $A \\times B , A \\otimes B , A \\wedge B$ 和 $[ A , B ]$。如果对一切 $A , B , C \\in { \\mathcal { A } }$. 都有 $A ( B C ) = ( A B ) C$，则称该代数为 <u id=\"concept-6-associative-algebra\" class=\"concept-term\" data-concept-identity=\"urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4\" data-concept-route=\"/concept-identities#concept-7e2e6da6-b92e-45a5-b490-c11370b027b4\">结合代数(associative algebra)</u>。如果对一切 $A$、$B \\in { \\mathcal { A } }$ 都有 $A B = B A$，则称其为 <u id=\"concept-6-commutative-algebra\" class=\"concept-term\" data-concept-identity=\"urn:uuid:f0e269d8-bae5-4032-9611-4e7c7b5f8518\" data-concept-route=\"/concept-identities#concept-f0e269d8-bae5-4032-9611-4e7c7b5f8518\">交换代数(commutative algebra)</u>。",
        "quote_sha256": "844ecd0d91de52f7e8930fee87b333f81e572bf8de8136423288117a6ff4d6d9",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "en/chapter-06/01-algebras-and-ideals",
          "lang": "en",
          "source_category": "original_transcription",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-9d656724251243d3eeae84c4"
    },
    {
      "concept_id": "urn:uuid:f0e269d8-bae5-4032-9611-4e7c7b5f8518",
      "book_id": "szekeres",
      "edition_id": "source-snapshot-bc3404156f6e",
      "lang": "en",
      "route": "en/chapter-06/01-algebras-and-ideals",
      "anchor": "first-5ae721da6d748fa5a23f-en",
      "chapter": "6",
      "section": "6.1",
      "offset": 2606,
      "range": [
        6855,
        7085
      ],
      "source_revision": "2358bd7fdfb406dfc4a25849c3a46b00cea749852ae0264ca4ab53cc328ad69e",
      "source_category": "original_transcription",
      "review_status": "unreviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "commutative",
      "context": {
        "range": [
          6021,
          7140
        ],
        "quote": "We have used capital letters $A$, $B$, etc. to denote vectors because algebras most frequently arise in spaces of linear operators over a vector space $V$. There is, however, nothing in principle to prevent the more usual notation $u$, $v , . .$. for vectors and to write $uv$ for their product. The vector product has been denoted by a simple juxtaposition of vectors, but other notations such as $A \\times B , A \\otimes B , A \\wedge B$ and $[ A , B ]$ may arise, depending upon the context. The algebra is said to be <u class=\"concept-term\" data-concept-identity=\"urn:uuid:7e2e6da6-b92e-45a5-b490-c11370b027b4\" data-concept-route=\"/concept-identities#concept-7e2e6da6-b92e-45a5-b490-c11370b027b4\" id=\"first-9f30c583afba7b647f72-en\">associative</u> if $A ( B C ) = ( A B ) C$ for all $A , B , C \\in { \\mathcal { A } }$. It is called <u class=\"concept-term\" data-concept-identity=\"urn:uuid:f0e269d8-bae5-4032-9611-4e7c7b5f8518\" data-concept-route=\"/concept-identities#concept-f0e269d8-bae5-4032-9611-4e7c7b5f8518\" id=\"first-5ae721da6d748fa5a23f-en\">commutative</u> if $A B = B A$ for all $A$, $B \\in { \\mathcal { A } }$",
        "quote_sha256": "22cc43778bc6a0e9c0783a68ffc2f283d527591f2f44c7d51a166783ab6d0538",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "zh/chapter-06/01-algebras-and-ideals",
          "lang": "zh-CN",
          "source_category": "translation",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-6f04ca82f6f9c77b10483080"
    }
  ],
  "registered_positions": [
    {
      "route": "en/chapter-06/01-algebras-and-ideals",
      "lang": "en",
      "section": "6.1",
      "signal": "term-mark",
      "label": "commutative",
      "english": "commutative",
      "offset": 2224
    },
    {
      "route": "zh/chapter-06/01-algebras-and-ideals",
      "lang": "zh-CN",
      "section": "6.1",
      "signal": "term-mark",
      "label": "交换代数",
      "english": "commutative algebra",
      "offset": 1722
    }
  ],
  "statements": [],
  "resolved_references": [],
  "reading_order": [
    "occ-9d656724251243d3eeae84c4",
    "occ-6f04ca82f6f9c77b10483080"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 2,
    "located_occurrences": 2,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "d6892a10df0eca1857ec9df7dcd7cd303e3d88d7b1aeeff441522d9ff7708491",
  "optional_enrichments": []
}